$ASA= \text{Angle-Side-Angle}$
I was wondering if $ASA$ still worked on triangles for the sphere. I have a pretty hard time visualizing triangles on the sphere because I know the sum of their interior angles can be more than $180^{\circ}$, which feels weird to me, since I am accustomed to triangles on the plane. I know $ASA$ works on triangles on the plane, as you can just find the third angle by subtracting the sum of the other angles from $180^{\circ}$, and then use the Law of Sines to find out the other side lengths. Would $ASA$ work on a triangle on a sphere? Thanks in advance.
Best Answer
Indeed. Knowing a side ($\gamma$) and the two angles at either side ($A$ and $B$), we can use the dual Law of Cosines to get the third angle ($C$): $$ \cos(C)=-\cos(A)\cos(B)+\sin(A)\sin(B)\cos(\gamma) $$ Then, we can use the Law of Sines as usual to get the other sides.
Alternatively, we can use the same dual Law of Cosines to get $$ \cos(\alpha)=\frac{\cos(A)+\cos(B)\cos(C)}{\sin(B)\sin(C)} $$ and $$ \cos(\beta)=\frac{\cos(B)+\cos(A)\cos(C)}{\sin(A)\sin(C)} $$ Note that, unlike plane trigonometry, we can determine a spherical triangle by AAA.